Among the variety of nanostructured materials, carbon allotropes play a
central role. The discovery of fullerene C\(_{60}\)  [1]} through laser
evaporation of graphite and its synthesis in macroscopic amounts [2]}
was triggered the study of many other carbon nanostructures, such as carbon
nanotubes [3]}, graphene and its derivatives [4]}.
Currently, carbon nanomaterials are promising materials for many
applications, and in particular for extreme nonlinear optics. Being the
member of the carbon allotropes, it is expected a strong HHG from fullerene
molecules. Experimentally, in Refs. [5]}, [6]} it is reported a
strong harmonic signal from C\(_{60}\)  plasma. Theoretical works predicted a
strong HHG from a C\(_{60}\)  molecule [7]}, [8]}, [9]} and solid C\(_{60}\)  [10]}. The theoretical analyses in Refs. [7]}, [8]}, [9]} are dominated by a single-particle picture, but it is
unclear how the electron-electron Coulomb interaction leaves its mark on HHG
and sub-cycle electronic response in these materials. Besides, in [7]}, moderately strong fields were considered, and excitonic lines were
termed as noninteger harmonics. However, these intrinsic exciton lines are
the result of Raman scattering of light, not harmonic radiation. Another
problem is how the symmetry groups of the most abundant fullerenes C\(_{60}\) 
and C\(_{70}\) , namely the icosahedron group and the dihedral group, affect
the HHG process in these materials.
